Cardinalities of Infinite Sets: Proving Bijective Relationship (0,1) and (0,2)

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Homework Help Overview

The discussion centers around the concept of cardinality in set theory, specifically examining the bijective relationship between the open intervals (0,1) and (0,2). The original poster attempts to understand why these two infinite sets can be considered to have the same cardinality.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the definition of "having the same cardinality" and its implications for both finite and infinite sets. The original poster questions the validity of their reasoning regarding bijective functions and cardinality in the context of infinite sets.

Discussion Status

Some participants have provided clarification on the definition of cardinality, noting that it applies to both finite and infinite sets. There is an acknowledgment of the original poster's correct application of the concept, although the discussion does not reach a definitive conclusion.

Contextual Notes

The original poster expresses uncertainty about the implications of bijective relationships for infinite sets, indicating a need for further exploration of the definitions involved.

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Homework Statement


Explain why ## (0,1 ) ## and ## (0,2)##

I have proved that a function ## f: (0,1 ) -> (0,2) ##defined by f(x) =2x is bijective.

Homework Equations


The Attempt at a Solution


I could state that due to the sets relationship being bijective they have the same cardinality, but this is only true if both sets are finite. Here, (0,1) and (0,2 ) are infinite.

Any help?
 
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What is your definition of "having the same cardinality"?
 
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Nevermind, you are right. The definition of same cardinality or numerically equivalency doesn't restrict finite from infinite sets.
 
knowLittle said:
Nevermind, you are right. The definition of same cardinality or numerically equivalency doesn't restrict finite from infinite sets.

Right! So you solved the problem correctly. Congratz!
 

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